GAMS/PATH User Guide Version 4.3

INITIAL POINT STATISTICS
Maximum of X. . . . . . . . . . 4.1279e+06 var: (w.29)
Maximum of F. . . . . . . . . . 2.2516e+00 eqn: (a1.33)
Maximum of Grad F . . . . . . . 6.7753e+06 eqn: (a1.29)
var: (x1.29)
INITIAL JACOBIAN NORM STATISTICS
Maximum Row Norm. . . . . . . . 9.4504e+06 eqn: (a2.29)
Minimum Row Norm. . . . . . . . 2.7680e-03 eqn: (g.10)
Maximum Column Norm . . . . . . 9.4504e+06 var: (x2.29)
Minimum Column Norm . . . . . . 1.3840e-03 var: (w.10)
Figure 3.6: PATH Output - Poorly Scaled Model
3.3.2 Poorly Scaled Models
Problems which are well-defined can have various numerical problems that
can impede the algorithm’s convergence. One particular problem is a badly
scaled Jacobian. In such cases, we can obtain a poor “Newton” direction
because of numerical problems introduced in the linear algebra performed.
This problem can also lead the code to a point from which it cannot recover.
The final model given to the solver should be scaled such that we avoid
numerical difficulties in the linear algebra. The output provided by PATH
can be used to iteratively refine the model so that we eventually end up with
a well-scaled problem. We note that we only calculate our scaling statistics at
the starting point provided. For nonlinear problems these statistics may not
be indicative of the overall scaling of the model. Model specific knowledge is
very important when we have a nonlinear problem because it can be used to
appropriately scale the model to achieve a desired result.
We look at the titan.gms model in MCPLIB, that has some scaling
problems. The relevant output from PATH for the original code is given in
Figure 3.6. The maximum row norm is defined as
max
and the minimum row norm is
1≤i≤n
1≤j≤n
min
1≤i≤n
1≤j≤n
| (∇F (x))ij |
| (∇F (x))ij | .
47